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Question 17293: Trace One Word under the Already Declared Mod-Count Language

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0033-17293 · q-20261014-advance-20261102-r07-regular-language-0043

Original question and givens

Use alphabet=Fin(2)={0,…,1} and the language whose number of occurrences of target_symbol=1 is congruent to residue=0 modulo modulus=19. Construct the complete DFA on states 0 through modulus−1, start 0 and accepting subset [residue]. A target symbol increments the state modulo the modulus; every other symbol leaves it unchanged. The invariant is state equals target-symbol count modulo the modulus for every prefix of every finite word, not just selected tests. No regex, NFA, epsilon move, minimization or pumping-lemma result is added. Word=[1, 0, 0, 0, 1]. Give (word,state_trace,final_state,accepted?), retaining the initial state and every transition.

  • G001 — Use alphabet=Fin(2)={0,…,1} and the language whose number of occurrences of target_symbol=1 is congruent to residue=0 modulo modulus=19. Construct the complete DFA on states 0 through modulus−1, start 0 and accepting subset [residue]. A target symbol increments the state modulo the modulus; every other symbol leaves it unchanged. The invariant is state equals target-symbol count modulo the modulus for every prefix of every finite word, not just selected tests. No regex, NFA, epsilon move, minimization or pumping-lemma result is added. Word=[1, 0, 0, 0, 1]. Give (word,state_trace,final_state,accepted?), retaining the initial state and every transition.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

mod-count state=0;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(0+0) mod19=0

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001

S002 · implies

Before

mod-count state=0;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(0+1) mod19=1

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S001

S003 · implies

Before

mod-count state=1;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(1+0) mod19=1

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S002

S004 · implies

Before

mod-count state=1;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(1+1) mod19=2

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S003

S005 · implies

Before

mod-count state=2;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(2+0) mod19=2

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S004

S006 · implies

Before

mod-count state=2;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(2+1) mod19=3

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S005

S007 · implies

Before

mod-count state=3;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(3+0) mod19=3

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S006

S008 · implies

Before

mod-count state=3;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(3+1) mod19=4

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S007

S009 · implies

Before

mod-count state=4;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(4+0) mod19=4

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S008

S010 · implies

Before

mod-count state=4;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(4+1) mod19=5

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S009

S011 · implies

Before

mod-count state=5;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(5+0) mod19=5

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S010

S012 · implies

Before

mod-count state=5;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(5+1) mod19=6

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S011

S013 · implies

Before

mod-count state=6;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(6+0) mod19=6

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S012

S014 · implies

Before

mod-count state=6;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(6+1) mod19=7

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S013

S015 · implies

Before

mod-count state=7;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(7+0) mod19=7

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S014

S016 · implies

Before

mod-count state=7;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(7+1) mod19=8

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S015

S017 · implies

Before

mod-count state=8;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(8+0) mod19=8

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S016

S018 · implies

Before

mod-count state=8;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(8+1) mod19=9

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S017

S019 · implies

Before

mod-count state=9;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(9+0) mod19=9

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S018

S020 · implies

Before

mod-count state=9;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(9+1) mod19=10

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S019

S021 · implies

Before

mod-count state=10;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(10+0) mod19=10

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S020

S022 · implies

Before

mod-count state=10;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(10+1) mod19=11

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S021

S023 · implies

Before

mod-count state=11;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(11+0) mod19=11

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S022

S024 · implies

Before

mod-count state=11;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(11+1) mod19=12

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S023

S025 · implies

Before

mod-count state=12;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(12+0) mod19=12

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S024

S026 · implies

Before

mod-count state=12;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(12+1) mod19=13

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S025

S027 · implies

Before

mod-count state=13;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(13+0) mod19=13

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S026

S028 · implies

Before

mod-count state=13;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(13+1) mod19=14

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S027

S029 · implies

Before

mod-count state=14;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(14+0) mod19=14

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S028

S030 · implies

Before

mod-count state=14;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(14+1) mod19=15

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S029

S031 · implies

Before

mod-count state=15;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(15+0) mod19=15

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S030

S032 · implies

Before

mod-count state=15;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(15+1) mod19=16

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S031

S033 · implies

Before

mod-count state=16;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(16+0) mod19=16

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S032

S034 · implies

Before

mod-count state=16;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(16+1) mod19=17

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S033

S035 · implies

Before

mod-count state=17;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(17+0) mod19=17

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S034

S036 · implies

Before

mod-count state=17;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(17+1) mod19=18

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S035

S037 · implies

Before

mod-count state=18;original symbol=0;target symbol=1;indicator=0;modulus=19

After

next state=(18+0) mod19=18

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S036

S038 · implies

Before

mod-count state=18;original symbol=1;target symbol=1;indicator=1;modulus=19

After

next state=(18+1) mod19=0

Conditions: none

Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S037

S039 · implies

Before

states Fin(19);complete typed table=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 0]];start0;accepting residue=0

After

empty prefix count=0 equals start;append any symbol adds exactly its target indicator to BOTH count and state modulo the modulus

Conditions: none

Both count and state start at0.Appending any symbol updates both by the same indicator,so induction preserves state equal to target-symbol count modulo m for every prefix of every word.

Step identifiers and dependencies

Rule: regular_mod_count_all_prefix_induction_base_step · Uses: G001, S038

S040 · implies

Before

for EVERY finite word,state equals target-symbol count mod19;accepting set=[0]

After

accepted iff count congruent to0;whole declared language has this complete DFA certificate

Conditions: none

Final-state membership in the specified residue characterizes the entire declared language.This uses the complete DFA and universal invariant,not finite tests,and adds no regex,epsilon moves,minimization or pumping lemma.

Step identifiers and dependencies

Rule: regular_mod_count_language_characterized_not_test_suite · Uses: G001, S039

S041 · implies

Before

selected mod-count word:original ordered word=[1, 0, 0, 0, 1];start=0

After

trace before consuming symbols=[0]

Conditions: none

Include the initial state.An empty word performs no transition;do not reorder symbols merely because a final answer might coincide.

Step identifiers and dependencies

Rule: regular_word_initial_state_empty_word_retained · Uses: G001, S040

S042 · implies

Before

selected mod-count word:word position=0;symbol=1;current state=0;complete original row=[0, 1]

After

next state=1;ordered prefix trace=[0, 1]

Conditions: none

Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.

Step identifiers and dependencies

Rule: regular_every_symbol_source_bound_transition · Uses: G001, S041

S043 · implies

Before

selected mod-count word:word position=1;symbol=0;current state=1;complete original row=[1, 2]

After

next state=1;ordered prefix trace=[0, 1, 1]

Conditions: none

Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.

Step identifiers and dependencies

Rule: regular_every_symbol_source_bound_transition · Uses: G001, S042

S044 · implies

Before

selected mod-count word:word position=2;symbol=0;current state=1;complete original row=[1, 2]

After

next state=1;ordered prefix trace=[0, 1, 1, 1]

Conditions: none

Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.

Step identifiers and dependencies

Rule: regular_every_symbol_source_bound_transition · Uses: G001, S043

S045 · implies

Before

selected mod-count word:word position=3;symbol=0;current state=1;complete original row=[1, 2]

After

next state=1;ordered prefix trace=[0, 1, 1, 1, 1]

Conditions: none

Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.

Step identifiers and dependencies

Rule: regular_every_symbol_source_bound_transition · Uses: G001, S044

S046 · implies

Before

selected mod-count word:word position=4;symbol=1;current state=1;complete original row=[1, 2]

After

next state=2;ordered prefix trace=[0, 1, 1, 1, 1, 2]

Conditions: none

Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.

Step identifiers and dependencies

Rule: regular_every_symbol_source_bound_transition · Uses: G001, S045

S047 · calculation

Before

selected mod-count word exact target-symbol indicator count := sum(1, 0, 0, 0, 1)

After

selected mod-count word exact target-symbol indicator count = 2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · implies

Before

selected mod-count word:original word=[1, 0, 0, 0, 1];exact target count=2;final state=2;accepting residue=0

After

accepted=No;final state equals count modulo19

Conditions: none

Independently count original-symbol indicators and cross-check the final trace state.An empty word has empty sum0 and no invented transition.

Step identifiers and dependencies

Rule: regular_selected_word_count_reverse_check · Uses: G001, S047

Reverse checks

  • S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S047 · premise_and_result_audit — Recompute selected mod-count word exact target-symbol indicator count with exact rational arithmetic, not rounded decimals.
  • S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([1, 0, 0, 0, 1], [0, 1, 1, 1, 1, 2], 2, No)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 2f56ee9b800050dc62cac661aec06ca5bc4e7911eb78d817034b07b0fe6b77e5
  • candidate record SHA-256: 2b23d65df37c048df5010b54e9dbc0051fb79b665706d55dd39dcd6328d8eded
  • semantic review receipt SHA-256: 4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3
  • question-first witness SHA-256: 22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON